Optimal. Leaf size=38 \[ \frac {x}{a}-\frac {\coth ^3(x) (1-\text {sech}(x))}{3 a}-\frac {\coth (x) (3-2 \text {sech}(x))}{3 a} \]
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Rubi [A] time = 0.09, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3888, 3882, 8} \[ \frac {x}{a}-\frac {\coth ^3(x) (1-\text {sech}(x))}{3 a}-\frac {\coth (x) (3-2 \text {sech}(x))}{3 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3882
Rule 3888
Rubi steps
\begin {align*} \int \frac {\coth ^2(x)}{a+a \text {sech}(x)} \, dx &=-\frac {\int \coth ^4(x) (-a+a \text {sech}(x)) \, dx}{a^2}\\ &=-\frac {\coth ^3(x) (1-\text {sech}(x))}{3 a}+\frac {\int \coth ^2(x) (3 a-2 a \text {sech}(x)) \, dx}{3 a^2}\\ &=-\frac {\coth (x) (3-2 \text {sech}(x))}{3 a}-\frac {\coth ^3(x) (1-\text {sech}(x))}{3 a}-\frac {\int -3 a \, dx}{3 a^2}\\ &=\frac {x}{a}-\frac {\coth (x) (3-2 \text {sech}(x))}{3 a}-\frac {\coth ^3(x) (1-\text {sech}(x))}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 33, normalized size = 0.87 \[ \frac {6 x-4 \tanh (x)-4 \coth (x)-2 \text {csch}(x)+6 x \text {sech}(x)}{6 a \text {sech}(x)+6 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 46, normalized size = 1.21 \[ -\frac {2 \, \cosh \relax (x)^{2} - {\left ({\left (3 \, x + 4\right )} \cosh \relax (x) + 3 \, x + 4\right )} \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} + \cosh \relax (x)}{3 \, {\left (a \cosh \relax (x) + a\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 40, normalized size = 1.05 \[ \frac {x}{a} - \frac {1}{2 \, a {\left (e^{x} - 1\right )}} + \frac {15 \, e^{\left (2 \, x\right )} + 24 \, e^{x} + 13}{6 \, a {\left (e^{x} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 56, normalized size = 1.47 \[ -\frac {\tanh ^{3}\left (\frac {x}{2}\right )}{12 a}-\frac {\tanh \left (\frac {x}{2}\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {1}{4 a \tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 47, normalized size = 1.24 \[ \frac {x}{a} - \frac {2 \, {\left (5 \, e^{\left (-x\right )} - 3 \, e^{\left (-3 \, x\right )} + 4\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 94, normalized size = 2.47 \[ \frac {\frac {5\,{\mathrm {e}}^{2\,x}}{6\,a}+\frac {5}{6\,a}+\frac {{\mathrm {e}}^x}{a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {\frac {1}{2\,a}+\frac {5\,{\mathrm {e}}^x}{6\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}+\frac {x}{a}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {5}{6\,a\,\left ({\mathrm {e}}^x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\coth ^{2}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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